Regularisation and validation of a depth-averaged model for granular flow consistent with the incompressible µ(I) rheology

We present a regularised extension of a three-equation depth-averaged model for granular flows recently derived by Deléage et al. (J. Fluid Mech., vol. 1009, 2025, p. A57). The previous model was obtained consistently with the µ(I)-rheology, with the help of an asymptotic expansion of the flow variables near a steady uniform flow. However, the range of slope at which steady uniform granular flows can be observed is limited, and the former model cannot be used outside this range, in particular for the liquid/solid and liquid/gas transitions. This is why we present in this paper an extension of the three-equation model, which is valid for any value of the slope. In order to achieve this, we introduce a regularisation coefficient enabling to model a transition between various flow regimes. This coefficient is written in a way that preserves the consistency with the µ(I)-rheology in the range of slope admitting steady uniform flows. Outside this range, the model keeps a well-posed structure. The predictions of the regularised model are then validated through comparisons with experimental data in various unsteady and unstationary regimes. We first study granular dry fronts, and show that the model successfully predicts the flattening of the profiles when the Froude number increases, while keeping a well defined front. We then show that the model provides accurate results for the final deposit left after a granular dam break on low inclination slopes. Finally, a third validation is performed in a configuration combining a steep slope and a variable topography.